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Maximal Regularity for Non-Autonomous Second Order
Cauchy Problems
Dominik Dier, El Maati Ouhabaz
To cite this version:
Dominik Dier, El Maati Ouhabaz. Maximal Regularity for Non-Autonomous Second Order Cauchy
Problems. Integr. Eqs and Op. Theory, 2014, 78 (3), pp.427-450. �10.1007/s00020-013-2109-6�.
�hal-00880886�
Maximal Regularity for Non-Autonomous Second
Order Cauchy Problems
Dominik Dier, El Maati Ouhabaz
∗November 8, 2013
Abstract
We consider non-autonomous wave equations
u¨(t) + B(t) ˙u(t) + A(t)u(t) = f (t) t-a.e.
u(0) = u0, ˙u(0) = u1.
where the operators A(t) and B(t) are associated with time-dependent sesquilinear forms a(t, ., .) and b defined on a Hilbert space H with the same domain V . The initial values satisfy u0∈ V and u1∈ H. We prove
well-posedness and maximal regularity for the solution both in the spaces
V′ and H. We apply the results to non-autonomous Robin-boundary
conditions and also use maximal regularity to solve a quasilinear problem.
Key words: Sesquilinear forms, non-autonomous evolution equations, maximal regularity, non-linear heat equations, wave equation.
MSC:35K90, 35K45, 35K92, 47F05.
1
Introduction
The present paper is a continuation of [ADLO13] which is devoted to maximal regularity for first order non-autonomous evolution equations governed by forms. Here we address the problem of maximal regularity for non-autonomous second order problems.
We consider Hilbert spaces H and V such that V is continuously embedded into
H and two families of sesquilinear forms
a: [0, T ] × V × V → C, b: [0, T ] × V × V → C
such that a(., u, v) : [0, T ] → C, b(., u, v): [0, T ] → C are measurable for all
u, v∈ V ,
|a(t, u, v)| ≤ MkukVkvkV (t ∈ [0, T ]),
and
Re a(t, u, u) + wkuk2H ≥ αkuk2V (u ∈ V, t ∈ [0, T ]) ∗Corresponding author.
where M ≥ 0, w ∈ R, and α > 0 are constants. We assume also that b satisfies the same properties. For fixed t ∈ [0, T ], we denote by A(t), B(t) ∈ L(V, V′
) the operators associated with the forms a(t, ., .) and b(t, ., .), respectively. Given a function f defined on [0, T ] with values either in H or in V′ and consider the
second order evolution equation (
¨
u(t) + B(t) ˙u(t) + A(t)u(t) = f(t) t-a.e.
u(0) = u0, ˙u(0) = u1.
(1.1)
with initial values u0∈ V and u1∈ H. This is a damped non-autonomous wave
equation. The equation without the factor ˙u, i.e., (
¨
u(t) + A(t)u(t) = f(t) t-a.e.
u(0) = u0, ˙u(0) = u1.
(1.2)
is a non-autonomous wave equation.
Our aim is to prove well-posedness and maximal regularity for (1.1) and (1.2). We shall prove three main results. The first one concerns maximal reg-ularity in V′ for the damped wave equation (1.1). We prove that for u
0 ∈ V, u1∈ H and f ∈ L2(0, T, V′) there exists a unique solution u ∈ H1(0, T, V ) ∩ H2(0, T, V′). This result was first proved by Lions [Lio61, p. 151] by assuming
regularity of t 7→ a(t, u, v) and t 7→ b(t, u, v) for every fixed u, v ∈ V . This regularity assumption was removed in Dautray-Lions [DL88, p. 667], but taking
f ∈ L2(0, T, H) and considering mainly symmetric forms. The general case was
given recently by Batty, Chill and Srivastava [BCS08] by reducing the problem to a first order non-autonomous equation. The result in [BCS08] is stated in the case u0= u1= 0, only. Our proof is different from [BCS08] and is inspired by
that of Lions [Lio61]. Next we consider maximal regularity in H. This is more delicate and needs extra properties on the forms a and b. We prove that if the forms are symmetric and t 7→ a(t, u, v) and t 7→ b(t, u, v) are piecewise Lipschitz on [0, T ] then for u0 ∈ V , u1 ∈ H and f ∈ L2(0, T, H) there exists a unique
solution u ∈ H1(0, T, V )∩H2(0, T, H) to the equation (1.1). We also allow some
non-symmetric perturbations of a and b. The third result (Theorem 5.1) con-cerns the wave equation (1.2). We prove that if a is symmetric and t 7→ a(t, u, v) is Lipschitz on [0, T ], then for every u0∈ V , u1 ∈ H and f ∈ L2(0, T, H) there
exists a unique solution u ∈ L2(0, T, V )∩H1(0, T, H)∩H2(0, T, V′) to the
equa-tion (1.2). This result is not new and was already proved by Lions [Lio61, p. 150] for the case u0= 0 and later in [DL88, p. 666] for u0∈ V and u1∈ H.
The-orem 5.1 is stated in order to have a complete picture of maximal regularity for wave equations with or without damping. The proof in [DL88] uses a Galerkin method and sectorial approximation. The proofs of the three main theorems use a representation result of Lions (see Theorem 2.4 below) for a given sesquilinear form E acting on a product of a Hilbert and pre-Hilbert spaces H × V. In each case we have to define the appropriate spaces H, V and the form E to which we apply Theorem 2.4. This idea was already used in [Lio61]. Our choice of the spaces H, V and the form E allow us to sharpen and extend some results from [Lio61] and assume less regularity on t 7→ a(t, u, v) and t 7→ b(t, u, v).
We illustrate our abstract results by two examples. The first one is a linear damped wave equation with time dependent Robin boundary conditions. The second is a quasi-linear second order non-autonomous problem. The latter is
treated by a fixed point argument but the implementation of this classical idea uses heavily a priori estimates that follow from our maximal regularity results for linear equations.
Acknowledgment
Some ideas in this work germinated during a visit of the second named author at the University of Ulm in the framework of the Graduate School: Mathematical Analysis of Evolution, Information and Complexity financed by the Land Baden-Württemberg and during the visit of the first named author at the University of Bordeaux. Both authors thank Wolfgang Arendt for fruitful discussions on the non-autonomous maximal regularity.
D. Dier is a member of the DFG Graduate School 1100: Modeling, Analysis and Simulation in Economics.
The research of E. M. Ouhabaz is partly supported by the ANR project “Har-monic Analysis at its Boundaries”, ANR-12-BS01-0013-02.
2
Preliminaries
Throughout this paper, V and H are separable Hilbert spaces over the field K= C or R. The scalar products of H and V and the corresponding norms will be denoted by (. | .)H, (. | .)V, k.kHand k.kV, respectively. We denote by V′ the
antidual of V when K = C and the dual when K = R. The duality between V′
and V is denoted by h., .i. Then hu, vi = (u | v)H for u ∈ H and v ∈ V .
We assume that
V ֒→
d H;
i.e., V is a dense subspace of H such that for some constant cH>0,
kukH≤ cHkukV (u ∈ V ). (2.1)
By duality and density of V in H one has
H ֒→
d V
′ .
The space H is then identified with a dense subspace of V′
(associating to u ∈ H the antilinear map v 7→ (u | v)H = hu, vi for v ∈ V ).
Let
a: [0, T ] × V × V → K be a family of sesquilinear and V -bounded forms; i.e.
|a(t, u, v)| ≤ MkukVkvkV (u, v ∈ V, t ∈ [0, T ]) (2.2)
for some constant M , such that a(., u, v) is measurable for all u, v ∈ V . We shall call a satisfying the above properties a V -bounded non-autonomous sesquilinear
form. Moreover we say that a is quasi-coercive if there exist constants α > 0, ω∈ R such that
Re a(t, u, u) + ωkuk2H≥ αkuk
2
V (u ∈ V, t ∈ [0, T ]). (2.3)
For t ∈ [0, T ], a V -bounded and quasi-coercive sesquilinear form a(t, ., .) is closed. The operator A(t) ∈ L(V, V′
) associated with a(t, ., .) is defined by hA(t)u, vi = a(t, u, v) for u, v ∈ V. (2.4) We may also associate with a(t, ., .) an operator on H by taking the part A(t) of A(t) on H; i.e.,
D(A(t)) := {u ∈ V : A(t)u ∈ H}
A(t)u := A(t)u. Note that if a(t, ., .) is symmetric, i.e.,
a(t, u, v) = a(t, v, u) for all u, v ∈ V , then the operator A(t) is self-adjoint.
For a Hilbert space E we denote by L2(0, T, E) the L2-space on (0, T ) of
functions with values in E and by Hk(0, T, E) we denote the usual Sobolev
space of order k of functions on (0, T ) with values in E. For u ∈ H1(0, T ; E) we
denote the first derivative by ˙u and for u ∈ H2(0, T ; E) the second derivative
by ¨u.
We start with the following differentiation result.
Lemma 2.1. Let
a: [0, T ] × V × V → K
be a V -bounded, quasi-coercive non-autonomous form. Suppose that it is Lips-chitz with LipsLips-chitz constant ˙M, that is
|a(t, φ, ψ) − a(s, φ, ψ)| ≤ ˙M|t − s|kφkVkψkV, t, s∈ [0, T ] and φ, ψ ∈ V.
Let u, v ∈ H1(0, T ; V ). Then a(., u, v) ∈ W1,1(0, T ) and there exists a non-autonomous form ˙a which is V -bounded with constant ˙M such that
a(., u, v)˙ = a(., u, ˙v) + a(., ˙u, v) + ˙a(., u, v)
If additionally a is symmetric then
a(., u, u)˙ = 2 Re a(., u, ˙u) + ˙a(., u, u).
Note that for u, v ∈ V we have dtda(t, u, v) = ˙a(t, u, v) for a.e. t ∈ [0, T ].
This lemma is a consequence of the next two results.
Lemma 2.2. Let u ∈ H1(0, T ; V ) and v ∈ H1(0, T ; V′
). Then hv(.), u(.)i ∈
W1,1(0, T ) and
hv(.), u(.)i˙ = h ˙v(.), u(.)i + hv(.), ˙u(.)i.
Proof. By Fubini’s Theorem we have
Z t 0 h ˙v(s), u(s)i ds = Z t 0 D ˙v(s), u(0) + Z s 0 ˙u(r) drEds = hv(t), u(0)i − hv(0), u(0)i + Z t 0 Z s 0 h ˙v(s), ˙u(r)i dr ds
= hv(t), u(0)i − hv(0), u(0)i + Z t 0 Z t r h ˙v(s), ˙u(r)i ds dr = hv(t), u(0)i − hv(0), u(0)i + Z t 0 hv(t), ˙u(r)i − hv(r), ˙u(r)i dr = hv(t), u(t)i − hv(0), u(0)i − Z t 0 hv(r), ˙u(r)i dr. Thus hv(t), u(t)i = hv(0), u(0)i + Z t 0h ˙v(s), u(s)i ds + Z t 0 hv(s), ˙u(s)i ds
which proves the claim.
Proposition 2.3. Let S : [0, T ] → L(V, V′) be Lipschitz continuous. Then the following assertions hold.
a) There exists a bounded, strongly measurable function ˙S: [0, T ] → L(V, V′
)
such that
d
dtS(t)u = ˙S(t)u (u ∈ V )
for a.e. t ∈ [0, T ] and
k ˙S(t)kL(V,V′)≤ L (t ∈ [0, T ]) where L is the Lipschitz constant of S.
b) If u ∈ H1(0, T ; V ), then Su := S(.)u(.) ∈ H1(0, T ; V′) and
(Su)˙ = ˙S(.)u(.) + S(.) ˙u(.). (2.5) Proposition 2.3 is proved in [ADLO13]. Lemma 2.1 follows from (2.4), Lemma 2.2 and Proposition 2.3.
We shall need the following representation result due to Lions. See [Lio59, p. 156], [Lio61, p. 61] or [ADLO13].
Theorem 2.4 (Lions’ Representation Theorem). Let H be a Hilbert space, V
a pre-Hilbert space such that V ֒→ H. Let E : H × V → K be sesquilinear such that
1) for all w ∈ V, E(., w) is a continuous linear functional on H; 2) |E(w, w)| ≥ αkwk2V for all w ∈ V
for some α > 0. Let L ∈ V′
. Then there exists u ∈ H such that Lw= E(u, w)
3
Maximal Regularity for the Damped Wave
Equation in V
′Let H, V be Hilbert spaces such that V ֒→ H. We define the following maximald
regularity space
MR(V, V, V′) := L2(0, T, V ) ∩ H1(0, T ; V ) ∩ H2(0, T ; V′) = H1(0, T ; V ) ∩ H2(0, T ; V′).
Let a : [0, T ] × V × V → C and b: [0, T ] × V × V → C be non-autonomous
V-bounded and quasi-coercive sesquilinear forms. We denote by A(t) and B(t)
their associated operators in the sense of (2.4). The following is our first result.
Theorem 3.1. For every u0 ∈ V , u1∈ H and f ∈ L2(0, T ; V′), there exists a unique solution u ∈ MR(V, V, V′) of the non-autonomous second order Cauchy problem
( ¨
u(t) + B(t) ˙u(t) + A(t)u(t) = f(t) t-a.e.
u(0) = u0, ˙u(0) = u1.
(3.1)
Moreover there exists a constant C > 0 such that
kukMR(V,V,V′)≤ C
h
ku0kV + ku1kH+ kfkL2(0,T ;V′)
i
. (3.2) As mentioned in the introduction, this theorem was first proved by Lions [Lio61, p. 151] under an additional regularity assumption on t 7→ a(t, u, v) and
t7→ b(t, u, v). This regularity assumption was removed in Dautray-Lions [DL88, p. 667], but taking f ∈ L2(0, T, H) and considering mainly symmetric forms
(they allow some non-symmetric perturbations). Their proof is based on a Galerkin method. Another proof of Theorem 3.1 was given recently by Batty, Chill and Srivastava [BCS08] but they consider only the case u0= u1= 0. Our
proof is based on Theorem 2.4 and is in the spirit of Lions [Lio61]. It is different from the proofs in [DL88] and [BCS08].
A classical result of Lions says that
MR(V, V′
) := L2(0, T, V ) ∩ H1(0, T ; V′
) ֒→ C([0, T ]; H), (3.3) and also that for u ∈ MR(V, V′
) the function ku(.)k2H is in W1,1(0, T ) with
(kuk2H)˙ = 2 Reh ˙u, ui, (3.4)
see [Sho97, p. 106] and [DL88, p.570]. This implies that MR(V, V, V′
) ֒→
C([0, T ]; V ) ∩ C1([0, T ]; H). Thus for u ∈ MR(V, V, V′), both u(0) and ˙u(0)
make sense.
We start with the following basic lemma.
Lemma 3.2. For v ∈ H1(0, T ; V ) we have Z T 0 kv(t)k 2 V dt 1/2 ≤ T Z T 0 k ˙v(s)k 2 V ds 1/2 +√Tkv(0)kV.
Proof. Note that v(t) = v(0) +Rt 0 ˙v(s) ds, thus Z T 0 kv(t)k 2 V dt = Z T 0 v(0) + Z t 0 ˙v(s) ds v(t) V dt = Z T 0 Z T s ( ˙v(s) | v(t)) V dt ds + Z T 0 (v(0) | v(t)) V dt ≤ Z T 0 Z T 0 k ˙v(s)kVkv(t)kV dt ds + Z T 0 kv(0)kVkv(t)kV dt ≤ Z T 0 kv(t)kV dt Z T 0 k ˙v(s)kV ds + kv(0)kV ≤T Z T 0 kv(t)k 2 V dt 1/2 T Z T 0 k ˙v(s)k 2 V ds 1/2 + kv(0)kV .
Proof of Theorem 3.1. It suffices to show that there exists a unique solution in
the case where T < T0 and T0 > 0 is a constant that depends only on the
constants M, ω and α of (2.2) and (2.3). Indeed we can extend this solution to [0, T ] for any fixed T as follows. We write the interval [0, T ] as a finite union of sub-intervals [τi, τi+1], each has length less than T0. On each interval
[τi, τi+1] we have a unique solution ui with ui(τi) ∈ V , ˙ui(τi) ∈ H and ui ∈
MR(V, V, V′
) ֒→ C1([τ
i, τi+1]; H) ∩ C([τi, τi+1]; V ). On [τi+1, τi+2] we solve the
equation with ui+1(τ
i+1) = ui(τi+1) and ˙ui+1(τi+1) = ˙ui(τi+1). We define u on
[0, T ] by u = ui on [τ
i, τi+1] and check easily that u ∈ MR(V, V, V′) (on [0, T ])
is the unique solution to (3.1).
We prove existence of a solution in the case where
T < T0= min α2 M2, α √ 2M . (3.5)
Note that we may assume throughout this proof that the forms a and b are both coercive. Indeed, set v(t) = ewtu(t) then we have
¨
v(t) + B(t) ˙v(t) + A(t)v(t)
= ewt ¨u(t) + (B(t) + 2w) ˙u(t) + (A(t) + wB(t) + w2)u(t) . (3.6)
Since a and b are quasi-coercive, we may choose w large enough such that b+2w and a + wb + w2 are coercive. Note also that v ∈ MR(V, V, V′) if and only if u∈ MR(V, V, V′).
We define the Hilbert space H := H1(0, T ; V ) endowed with its usual norm
kukH:= kukH1(0,T ;V ) and the pre-Hilbert space
V := {v ∈ H2(0, T ; V ) : ˙v(T ) = 0}
with norm k.kV:= k.kH. Further we define the sesquilinear form E : H × V → C
by E(u, v) := − Z T 0 ( ˙u | ¨v) H dt + Z T 0 b(t, ˙u, ˙v) dt + Z T 0 a(t, u, ˙v) dt + a(0, u(0), v(0))
and for u0∈ V , u1∈ H and f ∈ L2(0, T ; V′) we define F : V → C by F(v) := Z T 0 hf, ˙vi dt + a(0, u 0, v(0)) + (u1| ˙v(0))H. We claim that 1) E(., v) ∈ H′ and F ∈ V′;
2) E is coercive; i.e., there exists a C > 0 such that |E(v, v)| ≥ Ckvk2H for
all v ∈ V.
Suppose for a moment that 1) and 2) are satisfied. Then we can apply Lions’s representation theorem (see Theorem 2.4) and obtain u ∈ H such that
E(u, v) = F (v) ∀ v ∈ V. (3.7) We show that u is a solution of (3.1).
Let ψ(t) ∈ D(0, T ) and w ∈ V and choose v(t) :=Rt
0ψ(s) ds w. It follows from (3.7) that − Z T 0 h ˙u(t), wi ˙ ψ(t)dt = Z T
0 hf(t) − B(t) ˙u − A(t)u(t), wiψ(t) dt.
This means that ˙u ∈ H1(0, T ; V′
), hence u ∈ MR(V, V, V′) and
¨
u(t) + B(t) ˙u(t) + A(t)u(t) = f(t) t-a.e. (3.8) in V′
. For general v ∈ V, we use again (3.7) and integration by parts to obtain
( ˙u(0) | ˙v(0))H+ Z T 0 h¨u, ˙vi dt + Z T 0 b(t, ˙u, ˙v) dt + Z T 0 a(t, u, ˙v) dt + a(0, u(0), v(0)) = Z T 0 hf, ˙vi dt + a(0, u 0, v(0)) + (u1| ˙v(0))H.
This equality together with (3.8) imply that
( ˙u(0) | ˙v(0))H+ a(0, u(0), v(0)) = a(0, u0, v(0)) + (u1| ˙v(0))H.
Since v ∈ V is arbitrary we obtain that u(0) = u0and ˙u(0) = u1. Therefore, u is
a solution of (3.1) on [0, T ] for T ≤ T0 and T0 is such that the above properties
1) and 2) are satisfied.
Now we return to 1) and 2). Property 1) is obvious. We show the coercivity property 2). Let v ∈ V. The equality d
dtk ˙v(t)k 2 H = 2 Re(¨v(t) | ˙v(t))H implies Z T 0 Re(¨v| ˙v)H dt = −1 2k ˙v(0)k 2 H. It follows that |E(v, v)| ≥ Re E(v, v) = 1 2k ˙v(0)k 2 H+ Z T 0 Re b(t, ˙v, ˙v) dt
+ Z T
0
Re a(t, v, ˙v) dt + Re a(0, v(0), v(0)). We use coercivity of b, a and V -boundedness of a to obtain
|E(v, v)| ≥ 12k ˙v(0)k2H+ α Z T 0 k ˙vk 2 V dt − M Z T 0 kvkVk ˙vkV dt + αkv(0)k 2 V.
Therefore, by Young’s inequality, we have
|E(v, v)| ≥ 12k ˙v(0)k2H+ α 2 Z T 0 k ˙vk 2 V dt − M2 2α Z T 0 kvk 2 V dt + αkv(0)k 2 V.
Next we apply Lemma 3.2 to obtain
|E(v, v)| ≥ α2 −M 2T2 α Z T 0 k ˙vk 2 V dt + α−M 2T α kv(0)k2V.
Now we use (3.5) and the fact that by Lemma 3.2,RT
0 kvk 2 V dt is dominated (up to a constant) byRT 0 k ˙vk 2 V dt + kv(0)k 2 V. We obtain 2).
Next we prove uniqueness. Suppose that u and v are two solutions of (3.1) which are in MR(V, V, V′
). Set w = u − v. Clearly w ∈ MR(V, V, V′) and
satisfies (in V′)
¨
w(t) + B(t) ˙w(t) + A(t)w(t) = 0, w(0) = 0, ˙w(0) = 0. We show that w = 0. For fixed t ∈ (0, T ] we have
Z t 0 Reh ¨ w,w˙i ds + Z t 0 Re b(s, ˙w,w˙) ds + Z t 0 Re a(s, w, ˙w) ds = 0. Using (3.4) we have Z t 0 Reh ¨ w,w˙i ds = 1 2 Z t 0 k ˙ wk2H˙ ds = 1 2k ˙w(t)k 2 H− 1 2k ˙w(0)k 2 H= 1 2k ˙w(t)k 2 H, and hence 0 = 1 2k ˙w(t)k 2 H+ Z t 0 Re b(s, ˙w,w˙) ds + Z t 0 Re a(s, w, ˙w) ds ≥ 12k ˙w(t)k2H+ α Z t 0 k ˙ wk2V ds − M Z t 0 kwkVk ˙ wkV ds.
Here we used coercivity of b and V -boundedness of a. Therefore, by Lemma 3.2, we have 0 ≥ 12k ˙w(t)k2H+ α Z t 0 k ˙ wk2V ds − M Z t 0 kwk 2 V ds 1/2Z t 0 k ˙ wk2V ds1/2 ≥ 12k ˙w(t)k2H+ (α − MT ) Z t 0 k ˙ wk2V ds.
By (3.5) we obtain that w = 0. This shows uniqueness.
Finally, in order to prove the apriori estimate (3.2), we consider the operator
S: V × H × L2(0, T, V′) 7→ MR(V, V, V′), (u0, u1, f) 7→ u.
This is a linear operator which is well defined thanks to the uniqueness of the solution u of (3.1). It is easy to see that S is a closed operator. Therefore it is continuous by the closed graph theorem. This gives (3.2).
The previous proof does not give any information on the constant C in (3.2). For small time T one can prove that C depends only on the constants of the forms. This observation will be needed in our application to a quasi-linear problem.
Proposition 3.3. If T > 0 is small enough, then the constant C in (3.2) depends only on the constants w, α, M and T .
Proof. Let u ∈ MR(V, V, V′
) be the solution of (3.1). For fixed t ∈ (0, T ] we have Z t 0 Rehf, ˙ui ds = Z t 0 Reh¨u, ˙ui ds + Z t 0 Re b(s, ˙u, ˙u) ds + Z t 0 Re a(s, u, ˙u) ds. Since by (3.4) Z t 0 Reh¨u, ˙ui ds = 1 2 Z t 0 k ˙uk 2 H˙ ds = 1 2k ˙u(t)k 2 H− 1 2k ˙u(0)k 2 H,
it follows by Young’s inequality that 1 α Z t 0 kfk 2 V′ ds + α 4 Z t 0 k ˙uk 2 V ds ≥ Z t 0 kfkV ′k ˙ukV ds ≥ Z t 0 Rehf, ˙ui ds =1 2k ˙u(t)k 2 H− 1 2k ˙u(0)k 2 H+ Z t 0 Re b(s, ˙u, ˙u) ds + Z t 0 Re a(s, u, ˙u) ds ≥ −12k ˙u(0)k2H+ α Z t 0 k ˙uk 2 V ds − M Z t 0 kukVk ˙ukV ds ≥ −1 2k ˙u(0)k 2 H+ 3α 4 Z t 0 k ˙uk 2 V ds − M2 α Z t 0 kuk 2 V ds.
Here we used coercivity of b and V -boundedness of a. Therefore, by Lemma 3.2, we have 1 α Z t 0 kfk 2 V′ ds + 1 2k ˙u(0)k 2 H≥ α 2 Z t 0 k ˙uk 2 V ds − M2 α Z t 0kuk 2 V ds ≥ α 2 − t2(2M2+ α) α Z t 0 k ˙uk 2 V ds − t 2M 2+ α α ku(0)k2V + 1 2 Z t 0 kuk 2 V ds. (3.9)
where we choose t such that α
2 > t2 (2M2 +α) α . Finally, since ¨
we obtain that k¨u(s)k2V′ ≤ 3kf(s)k 2 V′ + 3M k ˙u(s)k 2 V + 3M ku(s)k 2 V s-a.e.
This together with (3.9) ends the proof of the proposition when T is such that
α
2 >
T2(2M2+ α)
α .
4
Maximal Regularity for the Damped Wave
Equation in H
Let V, H be separable Hilbert spaces such that V ֒→
d H and let
a, b: [0, T ] × V × V → K
be closed non-autonomous sesquilinear forms on which we impose the following conditions. Each can be written as the sum of two non-autonomous forms a(t, u, v) = a1(t, u, v) + a2(t, u, v), b(t, u, v) = b1(t, u, v) + b2(t, u, v) u, v ∈ V where
a1, b1: [0, T ] × V × V → K satisfy the following assumptions
a) |a1(t, u, v)| ≤ MkukVkvkV for all u, v ∈ V , t ∈ [0, T ];
b) a1(t, u, u) ≥ αkuk2V for all u ∈ V , t ∈ [0, T ] with α > 0;
c) a1(t, u, v) = a1(t, v, u) for all u, v ∈ V , t ∈ [0, T ];
d) a1 is piecewise Lipschitz-continuous; i.e., there exist 0 = τ0 < τ1 <· · · < τn= T such that
|a1(t, u, v) − a1(s, u, v)| ≤ ˙M|t − s|kukVkvkV
for all u, v ∈ V, s, t ∈ [τi−1, τi], i ∈ {1, . . . , n},
and similarly for b1. Of course we may choose the same constants M, ˙M and α
for both forms a1and b1. We may also choose that same sub-intervals 0 = τ0< τ1<· · · < τn= T for both forms.
The non-autonomous forms
a2, b2: [0, T ] × V × V → K are measurable and satisfy
and similarly for b2.
Note that by Lemma 2.1, if c is a Lipschitz form on [0, T ], we may define its derivative ˙c(t, ., .) and we have
|˙c(t, u, v)| ≤ ˙MkukVkvkV, u, v∈ V (4.1)
for some constant ˙M. We shall use this estimate for c = a1 and for c = b1 on
sub-intervals of [0, T ] where these forms are supposed to be Lipschitz.
Let us denote by A(t) and B(t) the operators given by hA(t)u, vi = a(t, u, v) and hB(t)u, vi = b(t, u, v) for all u, v ∈ V .
As in the previous section we consider the damped wave equation. Here we study the maximal regularity property in H rather than in V′
. We introduce the maximal regularity space
MR(V, V, H) := H1(0, T ; V ) ∩ H2(0, T ; H).
We have
Theorem 4.1. Let a = a1+ a2and b = b1+ b2 be non-autonomous V -bounded and quasi-coercive forms satisfying the above properties a) − e). Then for every u0, u1∈ V and f ∈ L2(0, T ; H), there exists a unique solution u ∈ MR(V, V, H) of the non-autonomous second order Cauchy problem
( ¨
u(t) + B(t) ˙u(t) + A(t)u(t) = f(t) t-a.e.
u(0) = u0,˙u(0) = u1
(4.2)
Moreover ˙u(t) ∈ V for all t ∈ [0, T ].
For a related result see Lions [Lio61, p. 155]. However the result proved there is restricted to u1 = 0 and assumes f, f′ ∈ L2(0, T, H). Our proof resembles
that of Theorem 3.1 and uses similar ideas as in Lions [Lio61]. We use the following lemma for the proof of Theorem 4.1.
Lemma 4.2. Suppose that the forms a1 and b1 are Lipschitz continuous on
[0, T ]. Let v ∈ H2(0, T ; V ) and ǫ > 0. Then
(i) Z T 0 e−λtRe b1(t, ˙v, ¨v) dt = λ2 Z T 0 e−λtb1(t, ˙v, ˙v) dt −1 2 Z T 0 e−λt˙b1(t, ˙v, ˙v) dt +12e−λTb1(T, ˙v(T ), ˙v(T )) − 1 2b1(0, ˙v(0), ˙v(0)). (ii) Z T 0 e−λtRe a 1(t, v, ¨v) dt = λ2e−λTa1(T, v(T ), v(T )) −λ2a1(0, v(0), v(0)) + e−λTRe a1(T, v(T ), ˙v(T )) − Re a1(0, v(0), ˙v(0)) +λ22 Z T 0 e−λta1(t, v, v) dt −λ 2 Z T 0 e−λt˙a1(t, v, v) dt − Z T 0 e−λtRe ˙a1(t, v, ˙v) dt − Z T 0 e−λta1(t, ˙v, ˙v) dt. (iii) Z T 0 e−λtRe b 1(t, ˙v, ¨v) dt + Z T 0 e−λtRe a 1(t, v, ¨v) dt ≥ 1 2(αλ − 2 ˙M− 2M) Z T 0 e−λtk ˙vk2V dt
+ (λ2(αλ − ˙M) −M˙22) Z T 0 e−λtkvk2V dt +1 2e −λTh (α − ǫ)k ˙v(T )k2V + (λα − ˙ M2 ǫ )kv(T )k 2 V i −12b1(0, ˙v(0), ˙v(0)) − λ 2a1(0, v(0), v(0)) − Re a1(0, v(0), ˙v(0)). Proof. The proof of (i) and (ii) is based on Lemma 2.1 and the product rule.
Part (i) is a direct consequence of the formulae
e−λtb1(t, ˙v, ˙v)˙ = −λe−λtb1(t, ˙v, ˙v) + e−λt˙b1(t, ˙v, ˙v) + 2e−λtRe b1(t, ˙v, ¨v). For (ii) we first calculate the following derivatives
Re e−λta1(t, v, v)˙ = −λe−λta1(t, v, v) + e−λtRe ˙a1(t, v, v) + 2e−λtRe a 1(t, v, ˙v) Re e−λta 1(t, v, ˙v)˙ = −λe−λtRe a1(t, v, ˙v) + e−λtRe ˙a1(t, v, ˙v) + e−λta 1(t, ˙v, ˙v) + e−λtRe a1(t, v, ¨v)
then we multiply the first equation by λ
2 and add the second equation. Now (ii)
follows by integration over t from 0 to T .
For (iii) we add (i) and (ii) and use coercivity of a1, b1 and V -boundedness
of a1, ˙a1, b1, ˙b1. Thus Z T 0 e−λtRe b1(t, ˙v, ¨v) dt + Z T 0 e−λtRe a1(t, v, ¨v) dt ≥ 12(αλ − ˙M− 2M) Z T 0 e−λt k ˙vk2V dt +λ 2(αλ − ˙M) Z T 0 e−λtkvk2V dt − ˙M Z T 0 e−λtkvkVk ˙vkV dt +12e−λThα k ˙v(T )k2V + λαkv(T )k 2 V − 2Mkv(T )kVk ˙v(T )kV i −12b1(0, ˙v(0), ˙v(0)) − λ 2a1(0, v(0), v(0)) − Re a1(0, v(0), ˙v(0)).
We apply Young’s inequality and see that the last term is bounded from below by 1 2(αλ − 2 ˙M− 2M) Z T 0 e−λtk ˙vk2V dt + (λ2(αλ − ˙M) − ˙ M2 2 ) Z T 0 e−λtkvk2V dt +1 2e −λTh (α − ǫ)k ˙v(T )k2V + (λα − ˙ M2 ǫ )kv(T )k 2 V i −12b1(0, ˙v(0), ˙v(0)) −λ2a1(0, v(0), v(0)) − Re a1(0, v(0), ˙v(0)) for ǫ > 0.
Proof of Theorem 4.1. Uniqueness follows from Theorem 3.1 and we only need
to prove existence of a solution. As in the proof of Theorem 3.1 we may assume that the forms a and b are both coercive (see (3.6)).
1- Lipschitz-continuous forms. Suppose first that the forms a1 and b1 are
We define the Hilbert space
H := {u ∈ H2(0, T ; H) ∩ H1(0, T ; V ) : u(0), ˙u(0), ˙u(T ) ∈ V } with norm kukH given by
kuk2H:= k¨uk 2 L2 (0,T ;H)+ kuk 2 H1 (0,T ;V )+ ku(0)k 2 V + k ˙u(0)k 2 V + k ˙u(T )k 2 V
and the pre-Hilbert space V := H2(0, T ; V ) with norm k.k
V := k.kH. Next we
define the sesquilinear form E : H × V → C by
E(u, v) := Z T 0 e−λt(¨u | ¨v)H dt + Z T 0 e−λtb(t, ˙u, ¨v) dt +Z T 0 e−λta(t, u, ¨v) dt + η( ˙u(0) | ˙v(0))V + η(u(0) | v(0))V,
where λ and η are positive parameters. Later on, we will choose them to be large enough. For u0, u1∈ V and f ∈ L2(0, T ; H), we define F : V → C by
F(v) := Z T
0 e−λt
(f | ¨v)H dt + η(u1| ˙v(0))V + η(u0| v(0))V
We proceed as in the proof of Theorem 3.1. Suppose for a moment that 1) E(., v) ∈ H′
and F ∈ V′
;
2) E is coercive; i.e., there exists a C > 0 such that |E(v, v)| ≥ Ckvk2H for
all v ∈ V.
Then by Lions’s representation theorem there exists u ∈ H such that
E(u, v) = F (v) (4.3) for all v ∈ V. For arbitrary w ∈ V and ψ ∈ D(0, T ) we take v(t) =Rt
0
Rs
0 ψ(r) dr ds w.
It follows from (4.3) that ¨
u(t) + B(t) ˙u(t) + A(t)u(t) = f(t)
in L2(0, T ; V′
). This identity applied to (4.3) implies that
η( ˙u(0) | ˙v(0))V + η(u(0) | v(0))V = η(u1| ˙v(0))V + η(u0| v(0))V
for all v ∈ V. Hence u(0) = u0and ˙u(0) = u1. This means that u ∈ MR(V, V, H)
is a solution of (4.2).
It remain to prove properties 1) and 2). Again, 1) is obvious and we focus on 2). Let v ∈ V. For ǫ ∈ (0, α) set
R:= ηk ˙v(0)k2V + ηkv(0)k2V +12e−λTh (α − ǫ)k ˙v(T )k2V + (λα − ˙ M2 ǫ )kv(T )k 2 V i −12b1(0, ˙v(0), ˙v(0)) − λ 2a1(0, v(0), v(0)) − Re a1(0, v(0), ˙v(0)).
By the V -boundedness of a1and b1 we have
R≥ 12e−λTh(α − ǫ)k ˙v(T )k2V + (λα − M˙ǫ2)kv(T )k2Vi+ (η −M
2 )k ˙v(0)k
2
+ (η −λM2 )kv(0)k2V − Mk ˙v(0)kVkv(0)kV.
Young’s inequality yields
R≥ C1 h k ˙v(T )k2V + kv(T )k 2 V + k ˙v(0)k 2 V + kv(0)k 2 V i
for some C1>0 provided λ and η are sufficiently large. Now
Re E(v, v) = Z T 0 e−λtk¨vk2H dt + Z T 0 e−λtRe b1(t, ˙v, ¨v) dt + Z T 0 e−λtRe b2(t, ˙v, ¨v) dt + Z T 0 e−λtRe a1(t, v, ¨v) dt + Z T 0 e−λtRe a2(t, v, ¨v) dt + η( ˙v(0) | ˙v(0))V + η(v(0) | v(0))V.
We apply assertion (iii) of Lemma 4.2, it follows that
Re E(v, v) ≥ Z T 0 e−λtk¨vk2H dt + Z T 0 e−λtRe b2(t, ˙v, ¨v) dt + Z T 0 e−λtRe a2(t, v, ¨v) dt +1 2(αλ − 2 ˙M− 2M) Z T 0 e−λt k ˙vk2V dt +1 2(λ(αλ − ˙M) − ˙M) Z T 0 e−λtkvk2V dt + R.
Thus V -boundedness of a2 and b2and Young’s inequality yield
Re E(v, v) ≥ Ckvk2H,
for some C > 0 provided that λ and η are sufficiently large. This proves 2). Finally, we have seen that the unique solution u satisfies ˙u(T ) ∈ V but we may replace in the previous arguments [0, T ] by [0, t] for any fixed t ∈ (0, T ) and obtain ˙u(t) ∈ V .
2- Piecewise Lipschitz-continuous forms. Suppose now that the forms a1 and
b1satisfy assumption d). We may replace in the first step the interval [0, T ] by [τi−1, τi]. There exists a solution ui ∈ H1(τi−1, τi; V ) ∩ H2(τi−1, τi; H) of the
equation ¨
v(t) + B(t) ˙u(t) + A(t)u(t) = f(t) a.e. t ∈ [τi−1, τi],
with prescribed ui(τ
i−1), ˙ui(τi−1) in V . We also know from the previous step
that ui(τ
i), ˙ui(τi) ∈ V . Now we can solve the previous equation on [τi, τi+1] and
obtain a solution ui+1 such that ui+1(τ
i) = ui(τi) and ˙ui+1(τi) = ˙ui(τi). We
define u on [0, T ] by u = uion [τ
i−1, τi]. It is easy to check that u ∈ MR(V, V, H)
5
The Wave Equation
Let H, V be Hilbert spaces such that V ֒→ H. Suppose a: [0, T ] × V × V →d
C is a Lipschitz-continuous, symmetric, V-bounded and quasi-coercive non-autonomous form. We denote again by A(t) the operator associated with a(t) on V′
and by A(t) the part of A(t) in H. We introduce the maximal regularity space
MR(V, H, V′) := L2(0, T ; V ) ∩ H1(0, T ; H) ∩ H2(0, T ; V′) for the second order Cauchy problem. We have the following result.
Theorem 5.1. There exists a unique solution u ∈ MR(V, H, V′) of the
non-autonomous second order Cauchy problem
( ¨
u(t) + A(t)u(t) = f(t) t-a.e. u(0) = u0,˙u(0) = u1
(5.1)
for every u0 ∈ V , u1 ∈ H and f ∈ L2(0, T ; H). Moreover u(t) ∈ V for all t∈ [0, T ].
Note that by [DL88, p. 579], for every u ∈ MR(V, H, V′), ˙u can be viewed
as a continuous function from [0, T ] into the interpolation space (H, V′)
1 2. In
particular, ˙u(0) is well defined and ˙u(0) ∈ V′.
We start with the following lemma. Here ˙a(t, ., .) denotes the derivative of
t7→ a(t, ., .).
Lemma 5.2. Let v ∈ H2(0, T ; V ) with ˙v(T ) = 0. Then
(i) Z T 0 e−λtRe(¨v| ˙v)H dt = λ 2 Z T 0 e−λtk ˙vk2H dt −1 2k ˙v(0)k 2 H (ii) Z T 0 e−λtRe a(t, v, ˙v) dt = λ 2 Z T 0 e−λta(t, v, v) dt −1 2 Z T 0 e−λt˙a(t, v, v) dt +1 2a(T, v(T ), v(T )) − 1 2a(0, v(0), v(0))
Proof. For the first part we calculate the formula e−λtk ˙vk2H˙ = −λe−λtk ˙vk2H+ 2e−λtRe(¨v,˙v)H.
For (ii) we use Lemma 2.1 and the product rule to obtain
e−λta(t, v, v)˙ = −λe−λta(t, v, v) + 2e−λtRe a(t, v, ˙v) + e−λt˙a(t, v, v). Now the Lemma follows by integrating over t.
Proof of Theorem 5.1. First we prove existence of a solution. We define the
Hilbert space H := {u ∈ L2(0, T ; V ) ∩ H1(0, T ; H) : u(0), u(T ) ∈ V } with norm
kukHsuch that kuk 2 H:= kuk 2 L2(0,T ;V )+k ˙uk 2 L2(0,T ;H)+ku(0)k 2 V+ku(T )k 2 V and the
pre-Hilbert space V := {v ∈ H2(0, T ; V ) : ˙v(T ) = 0} with norm k.k
V := k.kH. Further we define E : H × V → C by E(u, v) := − Z T 0 ˙u (e −λt˙v)˙ H dt
+ Z T
0
e−λta(t, u, ˙v) dt + a(0, u(0), v(0))
and for u0∈ V , u1∈ H and f ∈ L2(0, T ; V′) we define F : V → C by
F(v) := Z T
0
e−λthf, ˙vi dt + a(0, u0, v(0)) + (u1| ˙v(0))H.
As in the previous sections, we use Lions’s representation Theorem. Suppose that the assumptions of Lions’s Theorem are satisfied. Then there exists a
u∈ H such that
E(u, v) = F (v) (5.2) for all v ∈ V. For the particular choice of v(t) := ψ(t)w where ψ ∈ D(0, T ) and
w∈ V we obtain from (5.2) that
Z T 0 h ˙u, wi(e −λtψ˙(t))˙ dt =Z T 0 hf − Au, wie −λtψ˙(t) dt.
This implies that ˙u ∈ H1(0, T ; V′
), hence u ∈ MR(V, H, V′
) and that ¨
u(t) + A(t)u(t) = f(t) t-a.e. (5.3) Following the proof of Lemma 1 and Theorem 2 in [DL88, p. 571 and 575] we can integrate by parts in the first term of E(u, v) to obtain
E(u, v) = h ˙u(0), ˙v(0)i + Z T 0 e−λth¨u, ˙vi dt + Z T 0 e−λta(t, u, ˙v) dt + a(0, u(0), v(0)) = Z T 0 e−λthf, ˙vi dt + a(0, u0, v(0)) + (u1| ˙v(0))H
where we used the identity (5.2). This together with (5.3) implies that h ˙u(0), ˙v(0)i + a(0, u(0), v(0)) = a(0, u0, v(0)) + (u1| ˙v(0))H.
Since v ∈ V was arbitrary this shows that u(0) = u0and ˙u(0) = u1.
Next we check the assumptions of Theorem 2.4. Assumption 1) is again easy to verify. Let v ∈ V, then integration by parts yields to
|E(v, v)| ≥ Re E(v, v) = k ˙v(0)k2H+ Z T 0 e−λtRe(¨v| ˙v)H dt + Z T 0 e−λtRe a(t, v, ˙v) dt + a(0, v(0), v(0)).
Thus Lemma 5.2 applied to the first and second integral and Young’s inequality shows that Re E(v, v) ≥ 12k ˙v(0)k2H+ λ 2 Z T 0 e−λtk ˙vk2H dt + λ 2 Z T 0 e−λta(t, v, v) dt
−12 Z T 0 e−λt˙a(t, v, v) dt + 1 2a(T, v(T ), v(T )) + 1 2a(0, v(0), v(0)) ≥ Ckvk2H
for some C > 0 if λ is large enough. Note that we can choose C depending only on the coercivity, V -boundedness, Lipschitz constant of the form and on T .
Uniqueness: Let u ∈ MR(V, H, V′) be a solution of (5.1) where f = 0 and u0 = u1 = 0. We have to show that u = 0. Fix r ∈ [0, T ] and define vr(t) :=
RT t 1[0,r]u(s) ds. Then vr∈ H 1(0, T ; V ) with v r(r) = 0 and ˙vr= −1[0,r]u. We obtain 0 = 2 Z T 0 Reh¨u, v ri dt + 2 Z T 0 Re a(t, u, vr) dt = 2 Z r 0 Z r t Reh¨u(t), u(s)i ds dt − 2 Z r 0 Re a(t, ˙vr, vr) dt = 2 Z r 0 Z s 0 Reh¨u(t), u(s)i dt ds − 2 Z r 0 Re a(t, ˙vr, vr) dt = 2 Z r 0 Reh Z s 0 ¨ u(t) dt, u(s)i ds − Z r 0 (a(t, vr, vr))˙− ˙a(t, vr, vr) dt = 2 Z r
0 Reh ˙u, ui ds + a(0, v
r(0), vr(0)) − Z r 0 ˙a(t, vr, vr) dt ≥ ku(r)k2H+ αkvr(0)k2V − ˙M Z r 0 kv rk2V dt. We set w(r) := vr(0) =R r 0u(s) ds ∈ L 2(0, T ; V ). Then w(r) − w(t) = v r(t) and αkw(r)k2V ≤ ˙M Z r 0 kw(r) − w(t)k 2 V dt ≤ 2r ˙Mkw(r)k 2 V + 2 ˙M Z r 0 kw(t)k 2 V dt.
Let 0 < r0 < 2 ˙αM and set Cr0 := α − 2r0M >˙ 0, then for every r ∈ [0, r0] we
have kw(r)k2V ≤ 2 ˙M C −1 r0 Z r 0 kw(t)k 2 V dt.
We conclude by Gronwall’s lemma that w(r) = 0 for all r ∈ [0, r0], hence u = 0
on [0, r0]. Now we may proceed inductively to obtain u = 0 on [0, T ].
Remark 5.3. If we add a V × H-bounded perturbation to a as in Section 4, we
can still prove existence in Theorem 5.1. But for the uniqueness we have to assume additionally that this perturbation is also H × V -bounded.
Remark 5.4. Let B(t) be bounded operators on H with kB(t)kL(H)≤ MB for
a.e. t ∈ [0, T ]. We consider the wave equation (
¨
u(t) + B(t) ˙u(t) + A(t)u(t) = f(t) t-a.e.
u(0) = u0, ˙u(0) = u1
(5.4)
Then for u0 ∈ V , u1 ∈ H and f ∈ L2(0, T, V′) there exists a solution u ∈ MR(V, H, V′
) to (5.4). The proof is the same as above, one has only to change
E(u, v) into E(u, v) := − Z T 0 ˙u (e −λt˙v)˙ H dt
+ Z T 0 e−λt(B(t) ˙u | ˙v)H dt + Z T 0 e−λta(t, u, ˙v) dt + a(0, u(0), v(0)).
The uniqueness of u is however not clear except if the map t 7→ B(t) is Lipschitz. If this later condition is satisfied one can use similar ideas as in [DL88, p. 686] to prove uniqueness. The proof for uniqueness in Theorem 5.1 is similar to that of [DL88, p. 673].
6
Applications
In this section we give applications of our results. We consider two problems, one is linear and the second one is quasi-linear.
I) Laplacian with time dependent Robin boundary conditions.
Let Ω be a bounded domain of Rd with Lipschitz boundary Γ. Denote by σ be
the (d − 1)-dimensional Hausdorff measure on Γ. Let
β1, β2: [0, T ] × Γ → R
be bounded measurable functions which are Lipschitz continuous w.r.t. the first variable, i.e.,
|βi(t, x) − βi(s, x)| ≤ M|t − s| (i = 1, 2) (6.1)
for some constant M and all t, s ∈ [0, T ], x ∈ Γ. We consider the symmetric forms a, b: [0, T ] × H1(Ω) × H1(Ω) → R defined by a(t, u, v) = Z Ω∇u∇v dx + Z Γ β1(t, .)uv dσ. (6.2) and b(t, u, v) = Z Ω∇u∇v dx + Z Γ β2(t, .)uv dσ. (6.3)
respectively. The forms a, b are H1(Ω)-bounded and quasi-coercive. The first
statement follows readily from the continuity of the trace operator and the boundedness of β. The second one is a consequence of the inequality
Z Γ|u| 2 dσ ≤ ǫkuk2H1 (Ω)+ cǫkuk 2 L2 (Ω), (6.4)
which is valid for all ǫ > 0 (cǫ is a constant depending on ǫ). Note that (6.4)
is a consequence of compactness of the trace as an operator from H1(Ω) into L2(Γ, dσ), see [Nec67, Chap. 2 § 6, Theorem 6.2].
Let A(t) be the operator associated with a(t, ., .) and B(t) the operator asso-ciated with b(t, ., .). Note that the part A(t) in H := L2(Ω) of A(t) is interpreted
as (minus) the Laplacian with time dependent Robin boundary conditions
Here we use the following weak definition of the normal derivative. Let v ∈
H1(Ω) such that ∆v ∈ L2(Ω). Let h ∈ L2(Γ, dσ). Then ∂νv = h by definition
ifR Ω∇v∇w + R Ω∆vw = R Γhwdσ for all w ∈ H
1(Ω). Based on this definition,
the domain of A(t) is the set
D(A(t)) = {v ∈ H1(Ω) : ∆v ∈ L2(Ω), ∂νv+ β1(t)v|Γ= 0},
and for v ∈ D(A(t)) the operator is given by A(t)v = −∆v.
Maximal regularity on H for the first order Cauchy problem associated with
A(t) was proved in [ADLO13]. Here we study the second order problem. By Theorem 4.1, the damped wave equation
¨ u(t) − ∆ ˙u(t) − ∆u(t) = f(t) u(0) = u0, ˙u(0) = u1∈ H1(Ω)
∂ν( ˙u(t) + u(t)) + β2(t, .) ˙u(t) + β1(t, .)u(t) = 0 on Γ
has a unique solution u ∈ MR(V, V, H) = H2(0, T ; L2(Ω)) ∩ H1(0, T ; H1(Ω))
whenever f ∈ L2(0, T, L2(Ω)).
Indeed, Theorem 4.1 implies that there exists u ∈ MR(V, V, V′
) with u(0) =
u0, ˙u(0) = u1and
(¨u, v)H+ b(t, ˙u, v) + a(t, u, v) = (f, v)H (6.5)
for all v ∈ V and all t ∈ [0, T ] \ N, where N is a Lebesgue null set. Let
t∈ [0, T ] \ N, then for the special choice v ∈ D(Ω) we obtain that (6.5) implies ¨
u(t) − ∆ ˙u(t) − ∆u(t) = f(t). This together with (6.5) and the above definition
of the normal derivative shows
∂ν( ˙u(t) + u(t)) + β2(t, .) ˙u(t) + β1(t, .)u(t) = 0 on Γ. II) A quasi-linear problem.
Let Ω be a bounded open set of Rdand let H be the real-valued Hilbert space
L2(Ω, dx) and V be a closed subspace of H1(Ω) which contains H1
0(Ω). If V 6= H01(Ω) we assume that Ω is a Lipschitz domain to ensure that the embedding
of V in H is compact. This latter property is always true for V = H1 0(Ω) for
any bounded domain Ω.
For g, h ∈ L2(0, T ; H) we define the forms a
g,h, bg,h: [0, T ] × V × V → R by ag,h(t, u, v) = d X k,j=1 Z Ω ajk(t, x, g, h)∂ku∂jv dx and bg,h(t, u, v) = d X k,j=1 Z Ω bjk(t, x, g, h)∂ku∂jv dx.
We assume that the coefficients ajk, bjk: [0, T ] × Ω × R × R → R are uniformly
bounded on [0, T ] × Ω × R × R by a constant M > 0 and satisfy the usual ellipticity condition d X k,j=1 ajk(t, x, y, z)ξkξj≥ η|ξ|2, d X k,j=1 bjk(t, x, y, z)ξkξj ≥ η|ξ|2
for a.e. (t, x) ∈ [0, T ] × Ω and all y, z ∈ R, ξ ∈ Rd. Here η > 0 is a constant. Moreover we assume that ajk(t, x, ., .), bjk(t, x, ., .) are continuous for a.e. (t, x).
We denote by Ag,h(t) and Bg,h(t) the associated operators.
Given u0 ∈ V , u1 ∈ H and f ∈ L2(0, T ; V′) the second order Cauchy
problem
( ¨
u(t) + Bg,h(t) ˙u(t) + Ag,h(t)u(t) = f (t) t-a.e.
u(0) = u0, ˙u(0) = u1
(6.6) has a unique solution ug,h∈ MR(V, V, V′) by Theorem 3.1. Moreover, by
Propo-sition 3.3 there exists C > 0 and 0 < T0≤ T depending only on M and η such
that the solution of (6.6) on [0, T0] satisfies the estimate
kug,hkMR T0(V,V,V′)≤ C h ku0kV + ku1kH+ kfkL2(0,T 0;V′) i . (6.7) Note that C and T0 are independent of g and h. We want to show that the
quasi-linear problem (
¨
u(t) + Bu, ˙u(t) ˙u(t) + Au, ˙u(t)u(t) = f (t) t-a.e.
u(0) = u0,˙u(0) = u1
(6.8)
has a solution u in MR(V, V, V′). We define the mapping S : H1(0, T ; H) → H1(0, T ; H) by Sg := ug, ˙g. Note that by (6.7) and the fact that C is independent
of g and h, Im(S) is a bounded subset of MR(V, V, V′). Moreover by
Aubin-Lions lemma, MR(V, V, V′) is compactly embedded into H1(0, T ; H). Therefore,
if S is continuous then we can apply Schauder’s fixed point theorem to obtain
u ∈ H1(0, T ; H) such that Su = u. Thus u is also in MR(V, V, V′
) and u ∈ Im(S). Hence u is a solution of (6.8).
It remains to prove that S is continuous. Let gn → g in H1(0, T ; H) and
set un := Sgn. Since a sequence converges to a fixed element u if and only
if each subsequence has a subsequence converging to u we may deliberately take subsequences. Since L2(0, T ; H) is isomorphic to L2((0, T ) × Ω) we may
assume (after taking a sub-sequence) that gn → g and ˙gn → ˙g for a.e. (t, x).
Furthermore since the sequence un is bounded in MR(V, V, V′) we may assume
(after taking a sub-sequence) that un → u in H1(0, T ; H) and un ⇀ u in
MR(V, V, V′). Hence a
jk(t, x, gn,˙gn) → ajk(t, x, g, ˙g) and bjk(t, x, gn,˙gn) →
bjk(t, x, g, ˙g) for a.e. (t, x). Now the equality un= Sgn means that
h¨un,viL2(0,T ;V′),L2(0,T ;V )+ d X j,k=1 (∂j˙un| bjk(t, x, gn,˙gn)∂kv)L2(0,T ;H) + d X j,k=1 (∂jun| ajk(t, x, gn,˙gn)∂kv)L2 (0,T ;H)= hf, viL2 (0,T ;V′),L2 (0,T ;V )
for all v ∈ L2(0, T ; V ) and u
n(0) = u0, ˙un(0) = u1. By the dominated
con-vergence theorem ajk(t, x, gn,˙gn)∂kv→ ajk(t, x, g, ˙g)∂kv in L2(0, T ; H).
More-over un ⇀ u in MR(V, V, V′) implies that ∂jun ⇀ ∂ju and ∂j˙un ⇀ ∂j˙u in
L2(0, T ; H). Thus taking the limit for n → ∞ yields
h¨u,viL2(0,T ;V′),L2(0,T ;V )+
d
X
j,k=1
+
d
X
j,k=1
(∂ju| ajk(t, x, g, ˙g)∂kv)L2(0,T ;H)= hf, viL2(0,T ;V′),L2(0,T ;V )
for all v ∈ L2(0, T ; V ) and u(0) = u
0, ˙u(0) = u1. Note that for the initial
condition we have used that MR(V, V, V′
) ֒→ C1([0, T ]; H) ∩ C([0, T ]; V ), see
(3.3). This is equivalent to Sg = u. Hence S is continuous.
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Dominik Dier, Institute of Applied Analysis, University of Ulm, 89069 Ulm,
Germany,
dominik.dier@uni-ulm.de
El Maati Ouhabaz, Institut de Mathématiques (IMB), Univ. Bordeaux, 351,
cours de la Libération, 33405 Talence cedex, France, Elmaati.Ouhabaz@math.u-bordeaux1.fr
